Treats the 1D signal as the displacement of a vibrating medium and reads its modal structure off the zero-crossings — the 1D analogue of how sand gathers on the nodal lines of a vibrating Chladni plate to reveal the plate's mode shapes.
The four metrics probe zero-crossing spacings with tools borrowed from random matrix theory and multiscale analysis. Random matrix theory predicts specific gap statistics for different classes of systems (Poisson for uncorrelated, Wigner surmise / GOE for level-repelling quantum systems), which lets us read the signal's dynamical class off its node statistics without assuming anything about amplitude. A multi-scale sweep then checks whether pattern complexity grows with frequency in the same way Chladni figures do on a physical plate.
Mean consecutive spacing ratio r = min(sₙ, sₙ₊₁) / max(sₙ, sₙ₊₁) of zero-crossing intervals. Borrowed directly from random matrix theory: Poisson (uncorrelated crossings) gives ⟨r⟩ ≈ 0.386, GOE / Wigner (level-repelling, as in many-body quantum systems) gives ⟨r⟩ ≈ 0.536, and perfectly regular nodes give r = 1. Periodic waveforms saturate at 1.0 (Sine Wave, Square Wave, Van der Pol, every logistic periodic window), chaos sits near GOE (Logistic Edge-of-Chaos at 1.0 reflects near-regularity of the window), and noisy/clustered signals sit well below. The most discriminating metric in the geometry (F=8.61).
log(1 + Fano factor) of zero-crossing intervals; Fano = variance / mean, so Poisson gives F=1 and bursty crossings drive it upward. Devil's Staircase (8.96), fBm Persistent (8.41), ETH/BTC Ratio (8.18), and Perlin Noise (7.89) all have heavy zero-crossing burstiness — long quiet stretches punctuated by rapid flips. Correlates strongly with Möbius-S³:phi_return_cv (+0.817), flagging the same clustering signature from a different angle.
Power-law exponent of zero-crossing count versus bandpass center frequency across six octave bands. Measures how pattern complexity grows with the "driving frequency" — the 1D analogue of how Chladni figures become more intricate at higher resonant modes. Weak discriminator (F=1.03, rank low) but picks out sources where the bandpassed structure genuinely scales across octaves versus signals whose spectra are concentrated in a narrow band.
Kolmogorov-Smirnov distance of positive/negative domain lengths from an exponential distribution. For Poisson crossings, domain lengths are exponential; structured signals deviate. Devil's Staircase (0.89), Temperature Drift (0.79), Rudin-Shapiro (0.73), LIGO Hanford/Livingston (0.72/0.67) all have strongly non-exponential domain distributions — clumped or bimodal — consistent with regime-switching or structured modulation. Discovered by ShinkaEvolve (chladni v1, direction #6).
| Source | Domain | Value |
|---|---|---|
| Speech "Zero" | speech | 1.0000 |
| Speech "Five" | speech | 1.0000 |
| Speech "Nine" | speech | 1.0000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Partition Function | number_theory | 0.0000 |
| Primes | number_theory | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Devil's Staircase | exotic | 0.8910 |
| Temperature Drift | climate | 0.7925 |
| Rudin-Shapiro | number_theory | 0.7288 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Minkowski Question Mark | exotic | 0.0000 |
| Primes | number_theory | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.5 (Period-4) | chaos | 2.0377 |
| Logistic r=3.74 (Period-5 Window) | chaos | 1.9719 |
| Logistic Edge-of-Chaos | chaos | 1.9655 |
| ··· | ||
| Damped Pendulum | motion | -0.6527 |
| Duffing Oscillator | chaos | -0.5770 |
| 2-Torus Quasiperiodic | chaos | -0.4942 |
| Source | Domain | Value |
|---|---|---|
| Devil's Staircase | exotic | 8.9598 |
| fBm (Persistent) | noise | 8.4143 |
| ETH/BTC Ratio | financial | 8.1796 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Logistic Edge-of-Chaos | chaos | 0.0000 |
| Minkowski Question Mark | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 1.0000 |
| Logistic r=3.5 (Period-4) | chaos | 1.0000 |
| Logistic Edge-of-Chaos | chaos | 1.0000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Minkowski Question Mark | exotic | 0.0000 |
| Primes | number_theory | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Takagi Function | exotic | 0.9957 |
| fBm (Persistent) | noise | 0.9955 |
| Riemann-Hardy-Littlewood | exotic | 0.9955 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0012 |
| Source | Domain | Value |
|---|---|---|
| Forest Fire | exotic | 0.6537 |
| Speech "Five" | speech | 0.6130 |
| Rössler Hyperchaos | chaos | 0.5348 |
| ··· | ||
| fBm (Antipersistent) | noise | 0.0000 |
| Brownian Walk | noise | 0.0000 |
| Constant 0xFF | noise | 0.0000 |
Chladni treats a signal as a vibrating string and asks: what are the statistics of its node points? The answer sorts the atlas cleanly by dynamical class. Periodic waveforms have regular spacings (nodal_gap_ratio → 1); chaotic attractors have GOE-like level repulsion; heavy-tailed clustered processes (financial ratios, 1/f noise, seismic data) show both elevated clustering and non-exponential domain lengths. The geometry is not a chaos detector per se — it is a gap-statistics detector that happens to place chaotic systems in a different part of the plane than periodic or stochastic ones.